The geometry based on the postulates or descriptions of Euclid. One
of the critical assumptions of the Euclidean geometry is given in
his fifth postulate: through a point not on a line, one and only one line
be drawn parallel to the given line. See also
→ non-Euclidean geometry.

Any of several geometries which do not follow the postulates and
results of Euclidean geometry. For example, in a non-Euclidean
geometry through a point several lines can be drawn parallel to
another line. Or, the sum of the interior angles of a triangle differs
from 180 degrees. According to Einstein's general relativity theory, gravity
distorts space into a non-Euclidean geometry.

A → non-Euclidean geometry in which there are no
→ parallel lines, and the sum of the → angles
of a → triangle is always greater than 180°.
Riemannian figures can be thought of as figures
constructed on a curved surface. The geometry is called elliptic because the section
formed by a plane that cuts the curved surface is an ellipse.